Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type

1.   Introduction

In this paper, we discuss the following boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type

where $0 < \alpha\leq1,$ ${^C D}^\alpha_{0^+}$ is Caputo fractional derivative, $f\in C(J\times\mathbb{R}, \mathbb{R}\setminus\{0\})$, $k\in C(J\times\mathbb{R}, \mathbb{R})$, $g\in \mathcal{C}(J\times\mathbb{R}, \mathbb{R})$, and $a, \; b, \; c$ are real constants with $a+b\neq0$.

Let $J = [0, T]$ be a bounded interval in $\mathbb{R}$ with $T > 0$. Let $C(J\times\mathbb{R}, \mathbb{R})$ denote the class of all continuous functions $k: J\times\mathbb{R}\to\mathbb{R}$. Let $\mathcal{C}(J\times\mathbb{R}, \mathbb{R})$ denote the class of functions $g: J\times\mathbb{R}\to\mathbb{R}$ such that

(ⅰ) the map $t\mapsto{\frac{u-k(t, u)}{f(t, u)}}$ is measurable for each $u\in\mathbb{R}$, and

(ⅱ) the map $u\mapsto{\frac{u-k(t, u)}{f(t, u)}}$ is continuous for each $t\in J$.

The class $\mathcal{C}(J\times\mathbb{R}, \mathbb{R})$ is called the Carathéodory class of functions on $J\times\mathbb{R}$ which are Lebesgue integrable when bounded by a Lebesgue integrable function on $J$.

By a solution of the boundary value problem (1.1), we mean a function $u$ such that

(ⅰ) the function $t\mapsto{\frac{u-k(t, u)}{f(t, u)}}$ is continuous for each $u\in\mathbb{R}$, and

(ⅱ) $u$ satisfies the equations in (1.1).

Fractional calculus has been drawn people's attention extensively. This is because of its extensive development by the theory and by its applications in various fields, such as physics, engineering, chemistry and biology; see ^[1]. Compared with integer derivatives, fractional derivatives are used for a better description of considered material properties, and the design of mathematical models by the differential equations of fractional order can be more accurately illustrated the characteristics of the real-world phenomena, such as the exothermic reactions model having constant heat source ^[2], the fractional SIRS-SI model describing the transmission of malaria disease ^[3], the fractional model of nonlinear wave-like equations ^[4], the fractional Biswas-Milovic model having Kerr and parabolic law nonlinearities ^[5] and the fractional-order chaotic and hyperchaotic systems ^[6,7,8]. Many papers about the solvability for fractional equations and systems have appeared; see ^{[9,10,11,12,13,14,15,16,17,18,19,20]}.

Benchohra et al. ^[17] investigated the existence of solutions for first order boundary value problems for fractional order differential equations

where $0 < \alpha < 1,$ ${^C D}^\alpha_{0^+}$ is the Caputo fractional derivative, $f:\; [0, T]\times\mathbb{R}\to\mathbb{R}$ is a continuous function, and $a, \; b, \; c$ are real constants with $a+b\neq0$.

In recent years, the theory of nonlinear differential equations with perturbations has been a hot research topic; see ^{[18,19,20,21,22]}. Dhage ^[22] discussed the following first order hybrid differential equation with mixed perturbations of the second type

where $[t_0, t_0+a]$ is a bounded interval in $\mathbb{R}$ for some $t_0, \ a\in\mathbb{R}$ with $a > 0$, $f\in C([t_0, t_0+a]\times\mathbb{R}, \mathbb{R}\setminus\{0\})$ and $k, \ g\in C([t_0, t_0+a]\times\mathbb{R}, \mathbb{R})$. They developed the theory of hybrid differential equations with mixed perturbations of the second type, and gave some original and interesting results.

As far as we know, there are no results for the boundary value problem (1.1) of nonlinear fractional differential equations with mixed perturbations of the second type. From the above works, we consider the solvability of the boundary value problem (1.1). An existence theorem for the boundary value problem (1.1) is given under mixed Lipschitz and Carathéodory conditions. Our results in this paper extend and improve some well-known results.

The paper is organized as follows: Section 2 gives some definitions and lemmas to prove our main results. Section 3 establishes an existence theorem for the boundary value problem (1.1) under mixed Lipschitz and Carathéodory conditions by the fixed point theorem in Banach algebra due to Dhage. Section 4 presents an example to illustrate the main results, which is followed by the conclusion in Section 5.

2.   Preliminaries

For the convenience of the reader, we give some background materials from fractional calculus theory to facilitate analysis of the boundary value problem (1.1). These materials can be found in the recent literature, see ^[23,24].

Definition 2.1. (^[24]) The Caputo fractional derivative of order $\alpha > 0$ of a continuous function $f:(0, +\infty)\to \mathbb{R}$ is given by

where $n$ is the smallest integer greater than or equal to $\alpha$, provided that the right side is pointwise defined on $(0, +\infty)$.

Definition 2.2. (^[24]) The Riemann-Liouville fractional integral of order $\alpha > 0$ of a function $f:(0, +\infty)\to \mathbb{R}$ is given by

provided that the right side is pointwise defined on $(0, +\infty)$.

From the definition of the Caputo derivative, we can obtain the following statement.

Lemma 2.1. (^[24]) Let $\alpha > 0$. If we assume $u\in C(0, 1)\cap L(0, 1),$ then the fractional differential equation

has $u(t) = c_0+c_1t+c_2t^2+\cdots+c_{n-1}t^{n-1}, \ c_i\in \mathbb{R}, ~i = 0, 1, 2, \cdots, n-1,$ as unique solutions, where $n$ is the smallest integer greater than or equal to $\alpha$.

Lemma 2.2. (^[24]) Assume that $u\in C^n [0, 1]$ with a fractional derivative of order $\alpha > 0$ that belongs to $C^n [0, 1]$. Then

for some $c_i\in \mathbb{R}, \ i = 0, 1, 2, \cdots, n-1,$ where $n$ is the smallest integer greater than or equal to $\alpha$.

The following fixed point theorem in Banach algebra due to Dhage ^[23] is useful in the proofs of our main results.

Lemma 2.3. (^[23]) Let $Q$ be a closed convex and bounded subset of the Banach space $P$ and let $A, \ C:\ P\to P$ and $B:\ Q\to P$ be three operators such that

(a) $A$ and $C$ are Lipschitz with the Lipschitz constants $\alpha$ and $\beta$ respectively,

(b) $B$ is compact and continuous,

(c) $u = AuBv+Cu$ for all $v\in Q\Rightarrow u\in Q$, and

(d) $\alpha M+\beta < 1$, where $M = \|B(Q)\| = \sup\{\|B(u)\|:\ u\in Q\}.$

Then the operator equation $AuBu+Cu = u$ has a solution in $Q$.

3.   Existence result

In this section, we discuss the existence results for boundary value problems (1.1).

We place the boundary value problem (1.1) in the space $C(J, \mathbb{R})$ of all continuous functions defined on $J$. $\|\cdot\|$ denotes a supremum norm in $C(J, \mathbb{R})$ by

and a multiplication "$\cdot$" in $C(J, \mathbb{R})$ by

for $u, \ v\in C(J, \mathbb{R}).$ Clearly $C(J, \mathbb{R})$ is a Banach algebra with respect to above norm and multiplication in it. $L^1 (J, \mathbb{R})$ denotes the space of Lebesgue integrable functions on $J$ equipped with the norm $\|\cdot\|_{L^1}$ defined by

We present the following hypotheses.

$(A_1)$ There exist constants $L_1 > 0$ and $L_2 > 0$ such that

and

for all $t\in J$ and $u, \; v\in \mathbb{R}.$ $(A_2)$ There exists a function $h\in L^1(J, \mathbb{R})$ such that

for all $u\in \mathbb{R}.$

Lemma 3.1. Suppose that $a, \; b, \; c$ are real constants with $a+b\neq0$. Then for any $v\in L^1(J, \mathbb{R})$, the function $u$ is a solution of the boundary value problem

and

if and only if $u$ satisfies the integral equation

Proof. Let $u$ be a solution of the problem (3.1) and (3.2). Applying the Riemann-Liouville fractional integral $I^\alpha_{0^+}$ on both sides of (3.1), by Lemma 2.2, then we obtain

for some $\tilde{c}\in \mathbb{R}.$ Consequently, the general solution of (3.1) is

Substituting $t = 0$ and $t = T$ in the above equality implies

By (3.2), then we have

that is

Therefore, (3.3) holds.

Conversely, suppose that $u$ satisfies the equation (3.3). Applying the Caputo fractional operator of the order $\alpha$ on both sides of (3.3), then (3.1) is satisfied. Thus, substituting $t = 0$ and $t = T$ in (3.1) implies

Then,

Hence, (3.2) also holds.

Now we will give the following existence theorem for the boundary value problem (1.1).

Theorem 3.1. Suppose that $(A_1)$ and $(A_2)$ hold. Furthermore, if

then the boundary value problem (1.1) has a solution defined on $J$.

Proof. Set $U = C(J, \mathbb{R})$ and define a subset $S$ of $U$ by

where

$F_0 = \sup_{t\in J}|f(t, 0)|$ and $K_0 = \sup_{t\in J}|k(t, 0)|$.

Clearly, $S$ is a closed, convex and bounded subset of the Banach space $U$. By Lemma 3.1, the boundary value problem (1.1) is equivalent to the nonlinear integral equation

Define three operators $A, \; C:\ U\to U$ and $B:\ S\to U$ by

and

Then the equation (3.5) is transformed into the operator equation as

Next, we prove the operators $A, \; B$ and $C$ satisfy all the conditions of Lemma 2.3.

Firstly, we prove that $A$ is a Lipschitz operator on $U$ with the Lipschitz constant $L_1$. Let $u, \; v\in U$. Then by $(A_1)$,

for all $t\in J$. Taking supremum over $t$, then we have

for all $u, \; v\in U.$ This shows that $A$ is a Lipschitz operator on $U$ with the Lipschitz constant $L_1$. Similarly, it can be implied that $C$ is also a Lipschitz operator on $U$ with the Lipschitz constant $L_2$.

Next, we prove $B$ is a compact and continuous operator on $S$ into $U$. Firstly, we prove $B$ is continuous on $S$. Let $\{u_n\}$ be a sequence in $S$ converging to a point $u\in S$. Then by the Lebesgue dominated convergence theorem,

for all $t\in J$. This shows that $B$ is a continuous operator on $S$.

Next we prove $B$ is a compact operator on $S.$ It is enough to show that $B(S)$ is a uniformly bounded and equicontinuous set in $U.$ On the one hand, let $u\in S$ be arbitrary. Then by $(A_2)$,

for all $t\in J.$ Taking supremum over $t,$

for all $u\in S.$ This shows that $B$ is uniformly bounded on $S.$

On the other hand, let $t_1, \; t_2\in J$. Then for any $u\in S,$ we get

where $p(t) = \int_{0}^{t} h(s)d s$. Since the function $p$ is continuous on compact $J$, it is uniformly continuous. Hence, for $\varepsilon > 0,$ there exists a $\delta > 0$ such that

for all $t_1, t_2\in J$ and $u\in S.$ This shows that $B(S)$ is an equicontinuous set in $U.$ Now the set $B(S)$ is uniformly bounded and equicontinuous set in $U$, so it is compact by Arzela-Ascoli Theorem. Thus, $B$ is a and compact operator on $S$.

Next, we show that $(c)$ of Lemma 2.3 is satisfied. Let $u\in U$ and $v\in S$ be arbitrary such that $u = AuBv+Cu.$ Then, by assumption $(A_1),$ we have

Thus, we get

Taking supremum over $t,$

This shows that $(c)$ of Lemma 2.3 is satisfied.

Finally, we obtain

and so,

Thus, all the conditions of Lemma 2.3 are satisfied and hence the operator equation $AuBu+Cu = u$ has a solution in $S.$ Therefore, the boundary value problem (1.1) has a solution defined on $J.$

Remark 3.1. Some existence results were given for the boundary value problem (1.1):

$(I)$ with $k\equiv0$, and $f\equiv1$ by Benchohra et al. in ^[17];

$(II)$ with $\alpha = 1$, $a = 1$, and $b = 0$ by Dhage in ^[22];

$(III)$ with $\alpha = 1$, $k\equiv0$, and $f\equiv1$ by Tisdell in ^[25].

4.   An example

In this section, we will present an example to illustrate the main results.

Example 4.1 Consider the following boundary value problem

where $\alpha = \frac{1}{2}$, $T = 1$, $k(t, u(t)) = \frac{1}{8}\sin u(t)$, $f(t, u(t)) = \sqrt{u^2 (t)+1}$, $g(t, u(t)) = \cos u(t)$, $a = b = 1$ with $a+b\neq0$, and $c = \frac{1}{4}$.

Let $L_1 = 1$, $L_2 = \frac{1}{8}$, $h(t)\equiv1$. Then hypotheses $(A_1)$ and $(A_2)$ hold. Since

Hence, (3.4) holds. Therefore, by Theorem 3.1, the boundary value problem (4.1) has a solution.

5.   Conclusion

In this paper, we have studied the solvability for the boundary value problem (1.1) of nonlinear fractional differential equations with mixed perturbations of the second type. We have presented an existence theorem for the boundary value problem (1.1) of nonlinear fractional differential equations with mixed perturbations of the second type under mixed Lipschitz and Carathéodory conditions due to the fixed point theorem in Banach algebra due to Dhage. The main results have been well illustrated with the help of an example. Our results in this paper have been extended and improved some well-known results.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (61703180, 61803176, 61877028, 61807015, 61773010), the Natural Science Foundation of Shandong Province (ZR2019MF032, ZR2017BA010), the Project of Shandong Province Higher Educational Science and Technology Program (J18KA230, J17KA157), and the Scientific Research Foundation of University of Jinan (1008399, 160100101).

Conflict of interest

The authors declare that they have no competing interests in this paper.